We analyze a modification of the Richards growth model by introducing a time-dependent perturbation in the growth rate. This modification becomes effective at a special switching time, which represents the first-crossing-time of the Richards growth curve through a given constant boundary. The relevant features of the modified growth model are studied and compared with those of the original one. A sensitivity analysis on the switching time is also performed. Then, we define two different stochastic processes, i.e. a non-homogeneous linear birth-death process and a lognormal diffusion process, such that their means identify to the growth curve under investigation. For the diffusion process, we address the problem of parameters estimation through the maximum likelihood method. The estimates are obtained via meta-heuristic algorithms (namely, Simulated Annealing and Ant Lion Optimizer). A simulation study to validate the estimation procedure is also presented, together with a real application to oil production in France. Special attention is devoted to the approximation of switching time density, viewed as the first-passage-time density for the lognormal process.

翻译：本文通过引入生长率的时间依赖性扰动，对Richards生长模型进行修正。该修正在一个特定的切换时刻生效，该时刻表示Richards生长曲线首次穿越给定常数边界的时间。我们研究了修正后生长模型的相关特性，并与原始模型进行了比较。同时对切换时间进行了敏感性分析。随后，我们定义了两个不同的随机过程——非齐次线性生灭过程和对数正态扩散过程，使其均值与所研究的生长曲线一致。针对扩散过程，我们通过最大似然法解决了参数估计问题，并借助元启发式算法（即模拟退火算法和蚁狮优化器）获得估计值。本文还提出了验证估计程序的模拟研究，以及法国石油产量的实际应用案例。研究特别关注切换时间密度的近似计算，将其视为对数正态过程的首达时间密度。